| L(s) = 1 | + 9·3-s − 63.5·5-s − 49·7-s + 81·9-s − 743.·11-s + 606.·13-s − 572.·15-s − 1.45e3·17-s + 2.89e3·19-s − 441·21-s − 1.77e3·23-s + 914.·25-s + 729·27-s + 150.·29-s + 6.75e3·31-s − 6.69e3·33-s + 3.11e3·35-s − 5.55e3·37-s + 5.46e3·39-s + 1.54e4·41-s + 1.14e4·43-s − 5.14e3·45-s + 1.01e4·47-s + 2.40e3·49-s − 1.31e4·51-s − 1.35e4·53-s + 4.72e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·5-s − 0.377·7-s + 0.333·9-s − 1.85·11-s + 0.996·13-s − 0.656·15-s − 1.22·17-s + 1.84·19-s − 0.218·21-s − 0.697·23-s + 0.292·25-s + 0.192·27-s + 0.0332·29-s + 1.26·31-s − 1.06·33-s + 0.429·35-s − 0.666·37-s + 0.575·39-s + 1.43·41-s + 0.942·43-s − 0.378·45-s + 0.667·47-s + 0.142·49-s − 0.706·51-s − 0.664·53-s + 2.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.504003907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504003907\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 7 | \( 1 + 49T \) |
| good | 5 | \( 1 + 63.5T + 3.12e3T^{2} \) |
| 11 | \( 1 + 743.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 606.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.45e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.89e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.77e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 150.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.54e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.31e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76369386288637598233115183051, −9.796871795521251433747920260483, −8.643817319318985614139412850841, −7.902888211080406742560822979675, −7.22335309741655018471536186417, −5.80387098553843759783705087714, −4.52589599392285324292687284254, −3.47167453273810960455547794605, −2.50478223405585195365999899732, −0.63614273554842050961409129086,
0.63614273554842050961409129086, 2.50478223405585195365999899732, 3.47167453273810960455547794605, 4.52589599392285324292687284254, 5.80387098553843759783705087714, 7.22335309741655018471536186417, 7.902888211080406742560822979675, 8.643817319318985614139412850841, 9.796871795521251433747920260483, 10.76369386288637598233115183051
